Problem-Solving

One of the highest valued skills in modern society is analytical thinking. It's the ability to solve complex issues with problem-solving.

The issue is everyone thinks problem-solving skills are like talent – you're either born with it or you're not. Luckily for you and I, it's not. Problem-solving can be learned with the right approach and practice.

To analyze something is to break it down into its constituent parts. Consultants – business problem-solvers for hire – do this daily. Here's their approach to problem-solving.

The key to good problem-solving is breaking apart problems. Logic trees are a great tool for this. They look something like this.

The problem is broken up into chunks, which are then broken up into even smaller chunks, until the problem is manageable. A good example is a peanut butter and jelly sandwich. If we assume this sandwich is our problem, we can break it up into smaller parts; we have the peanut butter, the jelly, and the bread (both pieces). We could then further split up these elements. For instance, we could split up peanut butter into ground peanuts, salt, sugar, and many more ingredients.

When you break up these problems into issues, which are further split into sub-issues, we aim to have them be Mutually Exclusive, Collectively Exhaustive (MECE). Mutually exclusive means the problem is broken up into chunks that don't overlap with each other; they're separate entities, just like the jelly is a totally different component from the peanut butter. However, when you take the peanut butter, jelly, and bread together, they sum to the peanut sandwich, which is collectively exhaustive. The entire problem is represented by the respective components. While this is a more advanced topic, feel free to research it, as it will allow you to disaggregate problems in a more insightful way.

There's another very important, under-appreciated skill in analytical thinking and that's problem definition. It's a human tendency to fast forward to problem solving once the just of the problem is understood, but how many times has that resulted in a wrong answer on a test? If you're like me, that happens quite a bit.

To improve your problem-solving abilities, I'd suggest slowing down and making sure you fully comprehend the problem before solving it. Additionally, if it's a real-life problem and not the SAT, I'd suggest asking for clarification on certain words or misunderstandings. It's crucial to understand the problem fully before solving it. Otherwise, you're wasting value time. I can guarantee this approach will save you time in the long run, as well as lead to accurate solutions a higher percentage of the time.

I know what you're thinking. This is cool and all but how does this apply? I'm not willing to draw a logic tree to solve every math problem on the SAT or GMAT (for business school).

And you shouldn't. But we can still apply the same principle here.

Take this question from the GMAT. It's quite similar to a difficult SAT question requiring good problem-solving. But wait. We know that skill. So let's use it.

It's important to note that any problem can be divided into an infinite number of ways. However, I'd add that for most problems, only 5-10 approaches are logically different and there's usually 2-3 that make the most sense. In approaching the above problem, there are several ways to segment the problem. Here's the entire approach I'd take, using the problem-solving abilities from above. Keep in mind I'd never write down a logic tree on paper, but dividing the problem mentally in your head can improve your approach.

1) Fully understand the problem. The phrase "ordered pairs" appears critical, as well as "integers", so I make a mental note of this and ensure I understand their meaning.

2) Create a structure towards solving the problem. Here's how I'd lay this out:

If you run through the image above, you'll notice I've broken down the problem in terms of the two statements given. All the sub-issues (right most phrases) are obvious. This is the same approach most people take in their heads, but written out. Now that we have the equations broken down into their component parts, we can add to our tree.

In the above logic tree, I added insights using deductive logic – the same reasoning we all use to reach conclusions – to solve this problem. As you can see, the logic tree is useful in that it exposes different elements of the problem in a blatant manner, which can draw out new insights. The key insight in this problem is recognizing that n can only attain values that are multiples of 4 if the first statement is to sum to 20.

While the above problem may have been a bit redundant at times, I hope it illustrated that the key to effective problem-solving, after understanding the problem, is breaking the problem up into manageable pieces. There's several different ways to do this, and one or multiple segmentations may lead you to the answer.

And lastly, let's cover some ways to implement these techniques in everyday life, especially if the SAT isn't in your immediate future.

Everyday problems can be broken up into various components. Any kind of purchase can be broken up into its respective pieces quite easily and in several different ways. One could breakup a car purchase into: short-term (you can flex your new sports car to your friends) and long-term effects (you wouldn't be able to pay for your son's college), financial (price, gas, maintenance) vs non-financial (increased capacity, better for the environment, safety), a process approach (researching cars, test-driving, purchasing, everyday use, and maintenance), and beyond. The point is each is a valid method with different nuances; maybe looking at the non-financial, intangibles of a car purchase uncovers how safety is a key consideration you forgot and can add to the mix.

I'll leave you with one more interesting example. A recent consulting case I worked on required me to breakdown the spread of a virus – yes, this was case was created before COVID-19 – into manageable chunks. I broke it down using a process approach (reducing spread, improving diagnosis, and improving treatment), while a peer of mine broke it down in a more algebraic manner (population size times infection rate). Both approaches are valid and give different insight into how one might combat COVID-19.

Clearly, the applications of good problem-solving skills are endless, so I suggest you use start applying this knowledge wherever possible in the real world.